PROJECT SUMMARY This research project will develop statistical methods for the analysis of time-to-event, or failure time, data. Major areas of application include randomized controlled trials and epidemiologic cohort studies for the prevention or treatment of cancer or other diseases. The project aims to develop regression methods for the simultaneous analysis of multiple outcome variables in relation to treatments or exposures that may be evolving over the study follow-up period. The methods to be developed will be based on semiparametric regression models that include Cox models for marginal hazard functions and additive semiparametric regression models for pairwise and higher dimensional dependency functions. Using these models the failure time data will be characterized using a multivariate version of Dabrowska?s survivor function representation. A maximum likelihood approach, based on the probability distribution of the evolving failure time histories, will be used for parameter estimation. The work has potential to strengthen analyses of treatment effects, or regression effects more generally, for specific clinical outcomes by using data on other failure time outcomes to provide information censoring information. For example in a clinical trial with death as primary outcome, these methods will allow the occurrence of serious, but non-fatal, events during the study subject follow-up period to strengthen primary outcome treatment evaluations. The novel methods also will provide an efficient means of assessing the magnitude of dependencies among the risks for various outcome types, and their relationship to treatments or covariates. Many clinical trials or cohort study applications involve some form of cohort subsampling, with expensive biomarker values determined from raw materials (e.g., genomic measures from blood specimens) only for ?cases? that develop study diseases during cohort follow-up and corresponding ?controls? that do not. A second aim of this research project is to develop efficient analyses of treatment or covariate effects in the presence of cohort subsampling, for both univariate and multivariate failure time data. The methods development here will also rely on semiparametric maximum likelihood methods, with the novel aspect of including a nonparametric likelihood component for covariate history increments as they evolve over cohort follow-up. With univariate failure time data this work will lead to estimating functions for Cox model regression parameters and for observed covariate history parameters for iterative maximization, under nested case-control, case-cohort, or more general sampling schemes. Multivariate failure time extensions will combine semiparametric models for marginal hazard functions and for pairwise and higher dimensional dependency functions with completely nonparametric models for observed covariate histories. Asymptotic distributions for the novel estimation procedures will be developed using empirical process theory, and moderate sample properties will be evaluated using computer simulations, and using applications to Women?s Health Initiative and other datasets.